© The Institution of Engineering and Technology
Since the division with remainder cannot be implemented in multivariable polynomials, the twodimensional nonseparable wavelet transform cannot be lifted by using a similar way as that of univariate wavelet transforms. To solve this problem, a general lifting factoring method of twodimensional twochannel nonseparable stripe filter banks is presented. The constructing form of the polyphase matrices of the stripe filter banks is deduced and the general factoring of the polyphase matrices is given. Compared with the separable lifting wavelet transform, the proposed lifting factoring method can extract better texture information. The lifting form is more succinct than that of the tensor product lifting wavelet transform. The computation amount of the proposed factoring method for image decomposition is a quarter of the twodimensional twochannel nonseparable stripe filter bank and the original twodimensional twochannel nonseparable wavelet system is quickened. Moreover, the proposed lifting factorising method is faster than the traditional twodimensional twochannel nonseparable wavelet transform based on the Fourier transformation framework in which the size of each filter is greater than . The proposed lifting factorising method has better sparsity than that of the original wavelet transform and the famous twodimensional twochannel biorthogonal symmetric nonseparable wavelet transform.
References


1)

1. Sweldens, W.: ‘The lifting scheme: a construction of second generation wavelets’, SIAM J. Math. Anal., 1998, 29, (2), pp. 511–546.

2)

2. Daubechies, I., Sweldens, W.: ‘Factoring wavelet transform into lifting steps’, J. Fourier Anal. Appl., 1998, 4, (3), pp. 245–267.

3)

3. Kovačević, J., Vetterli, M.: ‘Nonseparable multidimensional perfect reconstruction filter bank and wavelet bases for Rn’, IEEE Trans. Inf. Theory, 1992, 38, (2), pp. 533–555.

4)

4. Chen, Q.H., Micchelli, C.A., Peng, S.L., et al: ‘Multivariate filter banks having matrix factorizations’, SIAM J. Matrix Anal. Appl., 2003, 25, (2), pp. 517–531.

5)

5. Muramatsu, S., Han, D., Kobayashi, T., et al: ‘Directional lapped orthogonal transform: theory and design’, IEEE Trans. Image Process., 2012, 21, (5), pp. 2434–2448.

6)

6. Muramatsu, S., Furuya, K., Yuki, N.: ‘Multidimensional nonseparable oversampled lapped transforms: theory and design’, IEEE Trans. Signal Process., 2017, 65, (5), pp. 1251–1264.

7)

7. Chai, Y., Li, H.F., Guo, M.Y.: ‘Multifocus image fusion scheme based on features of multiscale products and PCNN in lifting stationary wavelet domain’, Opt. Commun., 2011, 284, (5), pp. 1146–1158.

8)

8. Verma, V.S., Jha, R.K.: ‘Improved watermarking technique based on significant difference of lifting wavelet coefficients’, Signal Image Video Process., 2015, 9, (6), pp. 1443–1450.

9)

9. Quellec, G., Lamard, M., Cazuguel, G., et al: ‘Adaptive nonseparable wavelet transform via lifting and its application to contentbased image retrieval’, IEEE Trans. Image Process., 2010, 19, (1), pp. 25–35.

10)

10. Gouze, A., Antonini, M., Barlaud, M., et al: ‘Design of signaladapted multidimensional lifting scheme for lossy coding’, IEEE Trans. Image Process., 2004, 13, (12), pp. 1589–1603.

11)

11. Kaaniche, M., Benyahia, A.B., Popescu, B.P., et al: ‘Vector lifting schemes for stereo image coding’, IEEE Trans. Image Process., 2009, 18, (11), pp. 2463–2475.

12)

12. Li, H.L., Liu, G.Z., Zhang, Z.W.: ‘Optimization of integer wavelet transforms based on difference correlation structures’, IEEE Trans. Image Process., 2005, 14, (11), pp. 2463–2475.

13)

13. Piella, G., Heijmans, H.J.A.M.: ‘Adaptive lifting schemes with perfect reconstruction’, IEEE Trans. Signal Process., 2002, 50, (7), pp. 2463–2475.

14)

14. Kaaniche, M., Benyahia, A.B., Popescu, B.P., et al: ‘Nonseparable lifting scheme with adaptive update step for still and stereo image coding’, Signal Process., 2011, 91, (12), pp. 2767–2782.

15)

15. Hur, Y., Park, H.J., Zheng, F.: ‘MultiD wavelet filter bank design using Quillen–Suslin theorem for laurent polynomials’, IEEE Trans. Signal Process., 2014, 62, (20), pp. 5348–5358.

16)

16. Kovacevic, J., Sweldens, W.: ‘Wavelet families of increasing order in arbitrary dimensions’, IEEE Trans. Image Process., 2000, 9, (3), pp. 480–496.

17)

17. Vrankic, M., Sersic, D., Sucic, V.: ‘Adaptive 2D wavelet transform based on the lifting scheme with preserved vanishing moments’, IEEE Trans. Image Process., 2010, 19, (8), pp. 1987–2004.

18)

18. Gao, X.P., Xiao, F., Li, B.D.: ‘Construction of arbitrary dimensional biorthogonal multiwavelet using lifting scheme’, IEEE Trans. Image Process., 2009, 18, (5), pp. 942–955.

19)

19. Suzuki, T., Kudo, H.: ‘2D nonseparable blocklifting structure and its application to Mchannel perfect reconstruction filter banks for lossytolossless image coding’, IEEE Trans. Image Process., 2015, 24, (12), pp. 4943–4951.

20)

20. Mikhail, K.T., Woodburn, C.J.: ‘Factorization of MD polynomial matrices for design of MD multirate systems’. Electronic Proc. 15th Int. Symp. on the Mathematical Theory of Networks and Systems, University of Notre Dame, South Bend, Indiana, USA, 12–16 August 2002.

21)

21. Zhang, L., Makur, A., Xu, Z.M.: ‘On lifting factorization for 2D LPPRFB’. 2006 Int. Conf. on Image Processing, Atlanta, GA, USA, October 2006, pp. 2145–2148.

22)

22. Liu, B., Peng, J.X.: ‘Multispectral image fusion method based on two channels nonseparable wavelets’, Sci. China F, Inf. Sci., 2008, 51, (12), pp. 2022–2032.

23)

23. Horn, R.A., Johnson, C.R.: ‘Matrix analysis’ (Cambridge University Press, Cambridge, 1986), pp. 158–165.

24)

24. Wang, J.W., Chen, C.H., Chien, W.M., et al: ‘Texture classification using nonseparable twodimensional wavelets’, Pattern Recognit. Lett., 1998, 19, pp. 1225–1234.

25)

25. Gupta, A., Joshi, S.D.: ‘Twochannel nonseparable wavelets statistically matched to 2D images’, Signal Process., 2011, 91, (4), pp. 673–689.
http://iet.metastore.ingenta.com/content/journals/10.1049/ietipr.2017.0935
Related content
content/journals/10.1049/ietipr.2017.0935
pub_keyword,iet_inspecKeyword,pub_concept
6
6